Prove!

This is not a vector space. Exactly one of the axioms of a vector space is not satisfied.

Rom yes im sure the exercise say it's not.For this reason i can't solve the exercise

Guest "Exactly one of the axioms of a vector space is not satisfied"

Witch is this axiom? 

Ok.  Distribution of scalar mutliplication with respect to field addition doesn't hold.

we should have

$(a+b)\vec{v} = a \vec{v} \oplus b \vec{v}$

let's take a look

$(a+b)\vec{v} =\{(a+b)v_x,(a+b)v_y\}\\ a\vec{v} \oplus b \vec{v} =\left \{\sqrt[3]{a^3 v_x^3+b^3 v_x^3}, \sqrt[3]{a^3 v_y^3+b^3 v_y^3 }\right \}=\\ \left\{\sqrt[3]{a^3+b^3}v_x,~\sqrt[3]{a^3+b^3}v_y \right\}\\ \sqrt[3]{a^3+b^3} \neq a+b$

Thank you very much Rom!